q3_sol_18085_f06

# q3_sol_18085_f06 - MIT OpenCourseWare http/ocw.mit.edu...

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MIT OpenCourseWare http://ocw.mit.edu 18.085 Computational Science and Engineering I Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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18.085 Quiz 3 December 8, 2006 Professor Strang Your PRINTED name is: SOLUTIONS Grading 1 2 3 1) (30 pts.) (a) Suppose f ( x ) is a periodic function: f ( x ) = ± ² 0 for < x < 0 x e for 0 x f ( x + 2 n ) ikx Find the coeﬃcients c k in the complex Fourier series f ( x ) = c k e . What is c 0 ? What is | c k | 2 ? (b) Draw a graph of f ( x ) from 2 to 2 . Also draw a careful graph of df/dx . How quickly do the coeﬃcients of f ( x ) decay as k ± → and why ? (c) Find the Fourier coeﬃcients d k of df/dx . Do they approach a constant (or what pattern do they approach) as k ± → ? Explain the pattern from your graphs. for every integer n 1
Solution. (1+ ik ) x (1+ ik ) (a) c k = 1 e x e ikx dx = 1 e = 1 1 e = 1 ( 1) k e 2 0 2 (1 + ik ) 0 2 1 + ik 2 (1 + ik ) 2 c 0 = 1 e | c k | 2 = 1 ( e x ) 2 dx = 1 e 2 2 0 4 (b) The graph of f ( x ) includes a jump of 1 at x = 0 and a drop of e at x = . So df/dx x includes ± ( x ) e ± ( x

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## This note was uploaded on 04/12/2011 for the course MATH 18.085 taught by Professor Staff during the Fall '10 term at MIT.

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q3_sol_18085_f06 - MIT OpenCourseWare http/ocw.mit.edu...

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